(1)
For the deterministic automata functor \(TX = 2 \times X^A\) 
consider the Barr extension \(L = \overline{T}\) . Then elements of
\(T\mathcal {Q} X\)  are pairs \((b, (Y_a)_{a \in A})\) , where each \(Y_a\)  is a
subset of \(X\) , and
\(\nabla _X(b, (Y_a)_{a \in A})=\lbrace  (b, (x_a)_{a \in A}) \mid \forall a \in A : x_a \in Y_a\rbrace \qquad \text{for $b = 0, 1$}.\) 
The two Moss liftings
\(\lambda  ^0, \lambda  ^1: \mathcal {Q} ^A \rightarrow \mathcal {Q} (2 \times (-)^A)\) 
corresponding to the two \(|A|\) -ary operation symbols from the
presentation in ex:pres.1 are thus defined (slightly
abusing notation) by
\(\lambda  ^i((Y_a)_{a \in A}) = \lbrace (i, (x_a)_{a \in A}) \mid \forall a\in A : x_a \in Y_a\rbrace  \qquad \text{for $i = 0,1$}.\) 
(2)
As indicated in ex:pres, the finite powerset functor
\(\mathcal {P}_\omega  \)  has operations \(\tau  ^n/n\)  given by
\(\tau  ^n(x_1,\dots ,x_n)=\lbrace x_1,\dots ,x_n\rbrace \) . The Moss lifting
\(\lambda  ^n\)  associated to \(\tau  ^n\)  when using the Barr extension is
exactly the one given by (REF ) above.
(3)
Recall from ex:pres that the operations of the finite
distribution functor \(\mathcal {D} \)  take formal convex combinations. Via the
Barr extension, such an operation, determined by coefficients
\(p_1,\dots ,p_n\)  such that \(\sum p_i=1\) , induces the predicate
lifting \(\lambda  \)  given by \(\lambda  _X(A_1,\dots ,A_n)\)  consisting of
all \(\mu \in \mathcal {D} X\)  such that there exists a distribution on \(\in _X\)  (a
subset of \(X\times \mathcal {Q} (X)\) ) whose marginal distributions are \(\mu \) 
(on \(X\) ) and the distribution \(\nu \)  on \(\mathcal {Q} (X)\)  given by
\(\nu (\lbrace A_i\rbrace )=p_i\) , respectively. In fact, however, this description
can be substantially simplified; e.g. one readily checks that in
the case \(n=2\) , we actually have
\(\lambda  (A_1,A_2)=\lbrace \mu \in \mathcal {D} (X)\mid \mu (A_1)\ge p_1,\mu (A_2)\ge p_2,\mu (A_1\cup A_2)=1\rbrace .\) 
(The generalization to higher arities is via what is nowadays known
as the splitting lemma [1]}.)
(4)
For the finitary monotone neighbourhood functor \(\mathcal {M}_\omega  \) 
(Example ), we obtain Moss liftings as follows. Marti
and Venema [2]} define a diagonal-preserving lax
extension \(L\)  for \(\mathcal {M} \)  (which, then, restricts to \(\mathcal {M}_\omega  \) ) by
means of nested Egli-Milner liftings. An explicit description
of \(L\)  is
\(LR = \lbrace (\mathfrak {A},\mathfrak {B})\in \mathcal {M} X\times \mathcal {M} Y\mid \forall A\in \mathfrak {A}.\,R[A]\in \mathfrak {B},\forall B\in \mathfrak {B}.\,R^\circ  [B]\in \mathfrak {A} \rbrace \) 
for \(R\subseteq X\times Y\) . In particular, for \(\mathfrak {A} \in \mathcal {M} X\)  and
\(\Phi \in \mathcal {M} \mathcal {Q} X\subseteq \mathcal {Q} \mathcal {Q} \mathcal {Q} X\) , we have
\(\mathfrak {A} \in \nabla ^L_X(\Phi )\quad \text{iff}\quad \mathfrak {A} \mathbin {L(\in )} \Phi \quad \text{iff}\quad &\textstyle \forall \beta \in \Phi .\,\bigcup \beta \in \mathfrak {A} \text{ and }\\&\forall A\in \mathfrak {A}.\,\lbrace B\in \mathcal {Q} X\mid B\cap A\ne \emptyset \rbrace \in \Phi .\) 

Combining \(\nabla ^L\)  with the presentation of \(\mathcal {M}_\omega  \) 
(Example ) produces, for each choice of numbers
\(n\ge 0\)  and \(k_1,\dots ,k_n\ge 0\) , a \(\sum _{i=1}^n k_i\) -ary Moss
lifting \(\lambda  \)  given by
\(\lambda  ((A_{ij})_{i=1,\dots ,n;j=1,\dots ,k_i})=\lbrace \mathfrak {A} \in \mathcal {M}_\omega  X\mid \,&\textstyle \forall i.\,\bigcup _j A_{ij}\in \mathfrak {A} \text{ and }\\&\forall B\in \mathfrak {A}.\,\exists i.\,\forall j.\,B\cap A_{ij}\ne \emptyset \rbrace .\) 

Since \(\mathcal {M}_\omega  \)  preserves finite sets and the box modality \(\Box \)  as
described in Example  is separating, it is clear that
the Moss liftings are expressible using \(\Box \)  and Boolean
operators. Concretely, this works as follows. For readability, we
denote the predicate lifting interpreting \(\Box \)  by \(\Box \)  as well,
similarly for the dual modality \(\Diamond \) , so that
\(\Diamond _X(A):=\mathcal {M} X\setminus \Box _X(X\setminus A)=\lbrace \mathfrak {A} \in \mathcal {M} X\mid \forall B\in \mathfrak {A}.\,B\cap A\ne \emptyset \rbrace \) .
Then the Moss lifting \(\lambda  \)  as described above can be written as
\(\textstyle \lambda  ((A_{ij}))=\bigcap _i \Box _X(\bigcup _j A_{ij})\cap \bigcap _\pi \Diamond _X(\bigcup _i A_{i\pi (i)})\) 
where \(\pi \)  ranges over all selection functions assigning to each
\(i\in \lbrace 1,\dots ,n\rbrace \)  an index \(\pi (i)\in \lbrace 1,\dots ,k_i\rbrace \) .

